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Friday, March 25, 2016

The evolution of the expectation value of the volume v of the universe is plotted.
The blue and orange lines are the expanding and contracting branches in the
Wheeler-de Witt theory. The green curve follows from loop quantum cosmology
and exhibits a quantum bounce close to Planck density.

These days I have been working on lecture notes for an introductory lecture series on loop quantum gravity. An introductory article by Abhay Ashtekar introduced this subject via a discussion on loop quantum cosmology (LQC), where already many of the essential features of loop quantum gravity are present and can be studied in a simplified setting. I think that this is a very useful pedagogical approach and I also wanted to incorporate it into my lectures. I just finished my first draft of a lecture about this subject, focussing on a specific exactly soluble LQC model and its comparison to a similar quantisation using the Wheeler-de Witt framework. Mostly I follow the original paper, with some additional comments, slight rearrangements, and omission of more advanced material that is not necessary in an introductory course in my point of view. The current draft is available here, and comments are always welcome.

Tuesday, March 15, 2016

Changing the connection variables underlying loop quantum gravityalso changes the geometric operators. They measure geometry with respectto the metric encoded in the Yang-Mills electric field $E^a_i$.

tl;dr: In some cases there are such analogues, but they are rather awkward.

A comment to a recent post on the current status of the issue of the spectra of geometric operators in loop quantum gravity raised the question of whether such ambiguities can also be found in Yang-Mills theory. The question if of course very interesting, however I am not aware of any reference commenting on it. So let me try.

Friday, March 11, 2016

An example of non-locality in an observable algebra. Observables are
defined as fields at the endpoints of geodesics. Changing the metric
along the geodesic via $P^{rA}$, the field conjugate to some of the relevant
components of the spatial metric, leads to an apparent non-locality due to
a rerouting of the geodesic.

A recent post contained a talk of mine of a specific construction of observables in general relativity, where a physical coordinate system was specified by the endpoints of certain geodesics. The so constructed coordinates are simply Gaußian normal coordinates and the usual relational construction of observables as fields at a physical point could proceed in the standard manner.

The calculation which we performed illustrates nicely some points about the properties of observables which are often not spelled out, however seem of relevance for researchers interested in quantum gravity and should not be confused. In this post, we will gloss over global problems in defining observables, see e.g. this paper and references therein for a recent interesting discussion. See also this post for an earlier discussion of observables and the control we have on them.

1. Structure of the (sub-)algebra of observables
When we quantise, we always pick some preferred subalgebra of phase space functions that we want to quantise. It is well known that we cannot quantise all functions at the same time, as formalised in the Groenewold-van Hove theorem. Therefore, also in canonical quantum gravity such a choice has to be made, independently of whether we first quantise or first solve the constraints. We are interested here in the latter case, i.e. in the quantisation of a complete (= point separating on the reduced phase space) set of Dirac observables, or equivalently a complete set of phase space functions after gauge fixing the Hamiltonian and spatial diffeomorphism constraints, although similar statements should in principle also hold in the former case.
The precise nature of this choice will determine the properties of our observable algebra. In particular, examples can be given where

the resulting algebra has a perfectly local structure (e.g. when employing dust to specify a reference frame),

the resulting algebra is local, but some phase space functions not contained in the complete sub-algebra have non-local commutation relations

the resulting algebra is non-local

Roughly, the following happens. If we localise some field with respect to some locally defined structure, such as the values that four scalar fields take at a point, then the algebra remains local, where local means that Poisson brackets will be proportional to a delta-distribution in the coordinates defined by the scalar field coordinates. Another such example is to use dust. We can however also localise some field with respect to a more global structure, such as the endpoint of a spatial geodesic originating from some observer or the boundary of the spatial slice. Then, phase space functions with a non-vanishing Poisson bracket with components of the metric specifying this geodesic will have non-local (in the coordinates specified by the geodesics) Poisson brackets with fields localised at the endpoints of geodesics, just because they change the geodesic which defines the point where the field is evaluated. However, it is still possible to pick a complete set of such observables which have local Poisson brackets among themselves if we pay attention not to include fields not commuting with the components of the metric determining the path of the geodesics. This is in fact what is happening here. For the last case, we simply locate our fields with respect to some extended structure that is specified using non-commuting phase space functions, e.g. a spacetime geodesic. This doesn’t mean that locality breaks down, it simply means that we are considering observables which are smeared over regions of spacetime, and two such regions associated to different localisation points might overlap.

2. Locality of the physical Hamiltonian
Another question is whether the physical Hamiltonian that one derives with respect to a certain choice of time will be non-local. Again, in the case of dust as a reference field, it is local. However, already in the second case above one finds that is non-local, simply because it contains the phase space functions which have non-local Poisson brackets. Since a generic Hamiltonian will contain all fields of the theory, it seems that one has to avoid relational observables with respect to non-local structures if one wants to obtain a local Hamiltonian. However, it should be stressed again that one still might have a complete local observable algebra, even though the Hamiltonian is non-local. A detailed example is given here.

3. Independence of spacetime boundaries
All arguments made above are independent of the structure of the spacetime at infinity. In particular, the math involved in constructing observables and computing their algebra, see e.g. here, here, and here, does not refer to infinity. Also, the fact that the on-shell Hamiltonian constraint is a boundary term is not of relevance here, as it generates evolution via its Hamiltonian vector field, which is non-vanishing on-shell. Once a clock field is chosen on the other hand, one obtains a physical Hamiltonian with respect to that clock which is non-vanishing on-shell. Therefore, as long as one can locally define a clock, one can also have evolution.

Tuesday, March 8, 2016

This rerecorded talk was originally given at the spring meeting of the German Physical Society on February 29, 2016. It is based on this paper, where the algebra of a set of gravity observables constructed using a coordinate system specified by geodesics is computed. My interest in this question, next to general considerations in quantum gravity, comes from the AdS/CFT correspondence, where this construction has been used in the construction of the bulk / boundary dictionary. I plan to comment a little more on the content of the paper in the near future, highlighting some in my point of view important conceptual conclusions on the structure of general relativistic observable algebras.

Saturday, March 5, 2016

tl;dr: Sometimes yes, sometimes no, depending on how one passes to the physical Hilbert space.

The title for this post is shamelessly borrowed from this paper, where Bianca Dittrich and Thomas Thiemann have discussed this question some time ago, concluding that the answer could be negative at the level of the physical Hilbert space, i.e. after solving the Hamiltonian constraint. Carlo Rovelli then wrote a rebuttal, arguing that discreteness should remain also at the physical level. Since those papers, some time has passed and new light has been shed on this issue from various angles, to be reviewed in this post.

Thursday, March 3, 2016

From June 1-3, there will be a quantum gravity workshop in Helsinki, a kind of inaugural meeting for the field in Finnland. Mostly young people were invited to give talks, which should give a fresh view on the subject and hopefully result in collaboration across traditional subfield boundaries. I will talk about some recent ideas of connecting loop quantum gravity and string theory by using the gauge / gravity correspondence, with some comments already having appeared here.

On a related note, Sabine Hossenfelder has recently written a very nice article for quanta magazine about the possible connection between these subjects, based on interviews with several researchers in the field. Seeing that more people are becoming interested in this subject is certainly a great encouragement in further pursuing this direction.